Factorising quadratics, from x squared + bx + c to harder ones

What is a quadratic expression?

A quadratic expression is one where the highest power of x is x², such as x² + 7x + 12 or 2x² − 5x − 3. There's no x³ or higher. The number in front of x² is called its coefficient, and so is the number in front of x.

Children meet quadratics first when they expand double brackets at key stage 3: (x + 3)(x + 4) multiplies out to x² + 7x + 12. Factorising quadratics is that process run backwards, putting the two brackets back. It builds on factorising with a common factor, which puts back one bracket.

The key stage 4 programme of study covers Years 10 and 11 together and doesn't say which year each topic comes in, so ask the school for its scheme of work if you want to know when it's taught.

How to factorise quadratics of the form x squared + bx + c

To factorise x² + bx + c, find two numbers that multiply to make c and add to make b. Those two numbers go in the brackets.

1. List the factor pairs of c, the number on its own. The times table grid helps here.
2. Find the pair that adds to b, the number in front of x.
3. Write the brackets: (x + one number)(x + the other).
4. Check by expanding the brackets back out.

Worked example: factorise x squared + 7x + 12

The factor pairs of 12 are 1 and 12, 2 and 6, and 3 and 4. Only 3 and 4 add to make 7: 3 × 4 = 12 and 3 + 4 = 7.

So x² + 7x + 12 = (x + 3)(x + 4).

Check by expanding: x × x is x², then 4x and 3x make 7x, and 3 × 4 = 12. That's the question again. A second check, with x = 1: the question gives 1 + 7 + 12, which is 20, and the brackets give 4 × 5 = 20.

Worked example: factorise x squared + 2x − 15

When c is negative, one number in the pair is negative. You need two numbers that multiply to make −15 and add to make 2. Those are 5 and −3: 5 × −3 = −15, and 5 + −3 is 2.

So x² + 2x − 15 = (x + 5)(x − 3).

Check: expanding gives x² − 3x + 5x − 15, which is x² + 2x − 15.

Worked example: factorise x squared − 8x + 15

Now you need two numbers that multiply to make 15 and add to make −8. Two negatives multiply to make a positive, so the pair is −3 and −5.

So x² − 8x + 15 = (x − 3)(x − 5).

Check: expanding gives x² − 5x − 3x + 15, which is x² − 8x + 15.

Working out the signs

The signs follow a pattern, which comes from the rules for multiplying negative numbers:

• Both signs plus (+ bx + c): both numbers are positive. x² + 7x + 12 = (x + 3)(x + 4).
• Minus x term, plus number (− bx + c): both numbers are negative. x² − 8x + 15 = (x − 3)(x − 5).
• Minus number at the end (− c): one number is positive and one is negative, and the bigger one takes the sign of b. x² + 2x − 15 = (x + 5)(x − 3).

If there's no number on the end, like x² + 5x, take out x as a common factor instead: x(x + 5).

How to factorise a quadratic equation

You factorise the quadratic expression, then use the brackets to solve the equation. Strictly, you factorise an expression, not an equation, but the two go together. To solve x² + 7x + 12 = 0, factorise the left side to get (x + 3)(x + 4) = 0. If two things multiply to make 0, one of them must be 0, so x = −3 or x = −4. The guide to solving quadratic equations shows this step and the other methods.

Difference of two squares

A difference of two squares is a quadratic like x² − 25: one square take away another, with no x term. It factorises into two brackets that are the same except for the sign:

x² − 25 = (x + 5)(x − 5)

It's the same method as before, with b = 0. You need two numbers that multiply to make −25 and add to make 0, which are 5 and −5. Check by expanding: x² − 5x + 5x − 25. The two middle terms cancel out, which leaves x² − 25.

So x² − 49 = (x + 7)(x − 7), and x² − 1 = (x + 1)(x − 1). The difference of two squares is in the GCSE content for every student.

How to factorise a quadratic with a coefficient in front of x squared

Factorising harder quadratics, where the coefficient of x² isn't 1, such as 2x² + 7x + 3, is higher tier content: the DfE marks it for the more highly attaining students only. Schools teach more than one way to do it. A common one is to split the middle term:

1. Multiply the coefficient of x² by the number on its own.
2. Find two numbers that multiply to make that answer and add to make the coefficient of x.
3. Split the middle term into those two parts.
4. Factorise in pairs, taking a common factor out of the first two terms and then the last two.
5. Take out the matching bracket, and check by expanding.

Worked example: factorise 2x squared + 7x + 3

1. 2 × 3 = 6.
2. Two numbers that multiply to make 6 and add to make 7 are 6 and 1.
3. Split 7x into 6x + x: 2x² + 6x + x + 3.
4. Factorise each pair: 2x² + 6x is 2x(x + 3), and x + 3 is 1(x + 3).
5. Both parts have (x + 3), so the answer is (2x + 1)(x + 3).

Check: expanding gives 2x² + 6x + x + 3, which is 2x² + 7x + 3. With x = 1, the question gives 2 + 7 + 3, which is 12, and the brackets give 3 × 4 = 12.

Worked example: factorise 3x squared − 10x − 8

1. 3 × −8 is −24.
2. Two numbers that multiply to make −24 and add to make −10 are −12 and 2.
3. Split the middle term: 3x² − 12x + 2x − 8.
4. Factorise each pair: 3x(x − 4) + 2(x − 4).
5. The answer is (3x + 2)(x − 4).

Check: expanding gives 3x² − 12x + 2x − 8, which is 3x² − 10x − 8.

Look for a common factor first

Some quadratics with a number in front of x² have a common factor in every term. Take it out first, and what's left is often an easier quadratic. 2x² + 10x + 12 is 2(x² + 5x + 6), which is 2(x + 2)(x + 3).

Expanding and factorising quadratics: always check

Every answer can be checked by expanding it, because expanding and factorising are inverses. If expanding the brackets doesn't give the question back, something has gone wrong. Putting a small number such as x = 1 or x = 2 into both the question and the answer is a quick second check.

Common mistakes when factorising quadratics

• Finding a pair that multiplies but doesn't add. For x² + 7x + 12, 2 and 6 multiply to make 12, but add to make 8. Both conditions have to work.
• Getting the signs the wrong way round. (x − 5)(x + 3) expands to x² − 2x − 15, not x² + 2x − 15. Expanding catches this.
• Writing (x − 5)² for x² − 25. (x − 5)² is x² − 10x + 25. A difference of two squares has one plus bracket and one minus bracket.
• Forgetting a common factor. For 3x² + 6x, the answer is 3x(x + 2), not x(3x + 6).
• Using the simple method on a harder quadratic. With 2x² + 7x + 3, numbers that multiply to make 3 and add to make 7 don't exist. The coefficient of x² has to be included.

How to help at home

• Practise factor pairs. Quick questions like "two numbers that multiply to make 24 and add to make 10?" are the heart of factorising quadratics, and they come from knowing the times tables well.
• Ask for the check. Expanding the brackets afterwards takes a few seconds and catches most slips.
• Talk about the signs. Ask "should these numbers be positive or negative?" before your child writes the brackets.
• Keep it short. A few carefully checked questions do more than a long list done quickly.

Factorising quadratics questions to try

1. Factorise x² + 6x + 8.
2. Factorise x² + x − 12.
3. Factorise x² − 9x + 20.
4. Factorise x² − 36.
5. Factorise x² + 5x.
6. Higher tier: factorise 2x² + 11x + 5.

Answers

1. (x + 2)(x + 4), because 2 × 4 = 8 and 2 + 4 = 6
2. (x + 4)(x − 3), because 4 × −3 is −12 and 4 + −3 is 1
3. (x − 4)(x − 5), because −4 × −5 is 20 and −4 + −5 is −9
4. (x + 6)(x − 6), a difference of two squares
5. x(x + 5), taking out the common factor x
6. (2x + 1)(x + 5). 2 × 5 = 10, and 10 and 1 add to make 11, so split the middle term: 2x² + 10x + x + 5, which is 2x(x + 5) + 1(x + 5).

Factorising is also the first step in simplifying algebraic fractions. For the rest of GCSE maths, see the GCSE maths foundation and GCSE maths higher guides.